Unitary Categorical Symmetries

TL;DR

The paper extends the notion of unitary symmetry actions from invertible groups to non-invertible fusion category symmetries by promoting twisted sector local operators to $\ast$-representations of the tube algebra $\mathrm{Tube}(\mathcal{C})$ and establishing a $C^*$-structure via reflection positivity. It develops explicit tube-algebra constructions in two and three spacetime dimensions and classifies irreducible $\ast$-representations using higher $S$-matrices derived from the Symmetry TFT, identifying a correspondence with $\Omega^{D-2}(\mathcal{Z}^{\dagger}(\mathcal{C}))$. The work provides concrete examples including group symmetries, Tambara-Yamagami categories, Fibonacci categories in 2D, and 2-group symmetries in 3D, highlighting the role of central idempotents and induced representations. This framework yields a computable, unified approach to non-invertible symmetries and their action on both local and twisted sector observables, with potential extensions to boundary operators and higher-dimensional defects.

Abstract

Global invertible symmetries act unitarily on local observables or states of a quantum system. In this note, we aim to generalise this statement to non-invertible symmetries by considering unitary actions of higher fusion category symmetries $\mathcal{C}$ on twisted sector local operators. We propose that the latter transform in $\ast$-representations of the tube algebra associated to $\mathcal{C}$, which we introduce and classify using the notion of higher $S$-matrices of higher braided fusion categories.